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Mirrors > Home > MPE Home > Th. List > dyadval | Structured version Visualization version GIF version |
Description: Value of the dyadic rational function 𝐹. (Contributed by Mario Carneiro, 26-Mar-2015.) |
Ref | Expression |
---|---|
dyadmbl.1 | ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
Ref | Expression |
---|---|
dyadval | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
2 | oveq2 7163 | . . . 4 ⊢ (𝑦 = 𝐵 → (2↑𝑦) = (2↑𝐵)) | |
3 | 1, 2 | oveqan12d 7174 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 / (2↑𝑦)) = (𝐴 / (2↑𝐵))) |
4 | oveq1 7162 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1)) | |
5 | 4, 2 | oveqan12d 7174 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 + 1) / (2↑𝑦)) = ((𝐴 + 1) / (2↑𝐵))) |
6 | 3, 5 | opeq12d 4810 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉) |
7 | dyadmbl.1 | . 2 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) | |
8 | opex 5355 | . 2 ⊢ 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉 ∈ V | |
9 | 6, 7, 8 | ovmpoa 7304 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4572 (class class class)co 7155 ∈ cmpo 7157 1c1 10537 + caddc 10539 / cdiv 11296 2c2 11691 ℕ0cn0 11896 ℤcz 11980 ↑cexp 13428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 |
This theorem is referenced by: dyadovol 24193 dyadss 24194 dyaddisjlem 24195 dyadmaxlem 24197 opnmbllem 24201 opnmbllem0 34927 |
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